# How do you condense 3log_3x+4log_3y-4log_3z?

Dec 29, 2016

${\log}_{3} \left(\frac{{x}^{3} {y}^{4}}{{z}^{4}}\right)$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of logarithms}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log {x}^{n} \Leftrightarrow n \log x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow 3 {\log}_{3} x + 4 {\log}_{3} y - 4 {\log}_{3} z$

$= {\log}_{3} {x}^{3} + {\log}_{3} {y}^{4} - {\log}_{3} {z}^{4}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log x + \log y = \log \left(x y\right) , \log x - \log y = \log \left(\frac{x}{y}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow {\log}_{3} {x}^{3} + {\log}_{3} {y}^{4} = {\log}_{3} \left({x}^{3} {y}^{4}\right)$

$\Rightarrow {\log}_{3} \left({x}^{3} {y}^{4}\right) - {\log}_{3} {z}^{4} = {\log}_{3} \left(\frac{{x}^{3} {y}^{4}}{{z}^{4}}\right)$